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ORBIT SIZES and the DIHEDRAL GROUP of ORDER EIGHT By ORBIT SIZES AND THE DIHEDRAL GROUP OF ORDER EIGHT by Nathan A. Jones, B.S. A thesis submitted to the Graduate College of Texas State University in partial fulfillment of the requirements for the degree of Master of Science with a Major in Mathematics May 2018 Committee Members: Thomas Keller, Chair Anton Dochtermann Yong Yang COPYRIGHT by Nathan A. Jones 2018 FAIR USE AND AUTHOR'S PERMISSION STATEMENT Fair Use This work is protected by the Copyright Laws of the United States (Public Law 94- 553, section 107). Consistent with fair use as defined in the Copyright Laws, brief quotations from this material are allowed with proper acknowledgment. Use of this material for financial gain without the author's express written permission is not allowed. Duplication Permission As the copyright holder of this work I, Nathan A. Jones, authorize duplication of this work, in whole or in part, for educational or scholarly purposes only ACKNOWLEDGMENTS I would like to thank Dr. Thomas Keller for being my advisor for this thesis. From the start Dr. Keller helped find a great problem to tackle. His work inspired new ways of thinking. During the time we spent together each week he had unpar- alleled patience and encouragement. In addition he taught my first year Algebra where I got to learn the real beauty and elegance of finite group theory. I would also like to thank my committee members, Dr. Yong Yang, Dr. An- ton Dochtermann. Dr. Yang has always been incredibly friendly and encouraging in our encounters. Dr. Dochtermann was gracious and helpful in learning about and preparing this thesis. Finally, I would like to thank my dad who told me \there is no money in archeology, you'll starve", starting my journey of mathematical curiosity. iv TABLE OF CONTENTS Page ACKNOWLEDGEMENTS................................................................ iv CHAPTER I. INTRODUCTION.............................................................1 Definitions..................................................................1 Useful Theorems and Lemmas..........................................6 II. EXAMPLES AND STATEMENT OF RESULT.........................9 III. PROOF OF MAIN RESULT................................................ 11 A Reduction to Nilpotent Groups...................................... 11 A Reduction to p-groups................................................. 12 The Case Where V is Quasiprimitive.................................. 15 The Case Where V is Quasiprimitive.................................. 16 The Case Where D′ < G′ ................................................. 16 The case where D is nonabelian......................................... 18 The Case Where D is Abelian........................................... 19 The Case Where D has Exactly One Orbit Size M................. 19 The Case Where D has Exactly Two Orbits of Size M: ............ 20 The Case where D′ = G′ .................................................. 22 IV. The Open Case................................................................. 36 REFERENCES...................................................................... 37 I. INTRODUCTION Groups form one of the most basic structure available in abstract algebra, finding uses in a variety of areas in mathematics. In number theory the main object of study is the set of integers, a group. The solutions to polynomial equations give rise to a group. Well known results such as Euler's Theorem, Diophantine equa- tions, and solvability of polynomial equations have all benefited from the under- standing of groups and their structure [3, p.14]. A natural tool for studying groups is a group action on a set. Group actions provide a convenient way to study abstract groups. Seeing where a group can send a particular element of a set gives you a sense of that element's orbit. One question that arises is to study the sizes of these orbits. In this thesis we will be concerned with a lower bound for the largest orbit size. We will primarily concern ourselves with group actions where G is a finite nonabelian group acting on a finite faithful irreducible G-module V . It is known that SGS~SG′S, where G′ is the commutator sub- group, serves as a lower bound for the largest orbit size of G acting on V [8]. It is even known that this inequality is either strict, or equal and G is abelian, or there are at least two orbits of the largest orbit size in the action of G on V when G is nonabelian. The next natural step is to consider what happens if there are exactly two orbits of size SGS~SG′S in the action of G on V . In the cited paper the authors conjecture that this can only happen when the group is the dihedral group of order eight, while V = V (2; 3) is a two dimensional vector space over a field of 3 elements. In this thesis we will show that a slightly weaker version of this conjecture is in fact true. Definitions In this section we will formalize the terms used throughout this thesis. We 1 will also clarify notation which may vary in different sources. The definitions used in this paper have been taken from [6, 10] and the notation is consistent with litera- ture in finite group theory. This thesis will assume the reader is familiar with the basic concept of a group. A quick review of the properties of a group include a set G which is closed under a binary operation which is associative, contains an identity, and has in- verses. In this thesis all groups will be finite. That is to say the size of set G, de- noted SGS, is not infinite. An example of a group is the dihedral group on eight el- ements, denoted D8. This group consists of the four rotational and four ’flip’ sym- 2 3 2 3 metries of a square. These are represented by D8 = {1; r; r ; r ; s; sr; sr ; sr }, where r can be considered a 90 degree rotation of a square, and s can be considered a flip over a fixed axis of symmetry of a square. This produces the multiplication rule sr = r−1s. The first structures on a group G we will consider are smaller groups in- side G called subgroups. Subgroups provide a way to get an idea of the structure of larger groups. In this paper subgroups will be used to allow for induction on the 2 3 group. In the case of D8 we can look at only the set of rotations {1; r; r ; r } and see this forms a subgroup of D8. We will define some more special subgroups. Definition 1. A normal subgroup is a subgroup H ≤ G such that for all g ∈ G we have g−1Hg = H. This is denoted by H ⊴ G, and H ⊲ G if and only if H ⊴ G and H ≠ G. Notice that a normal subgroup is a group that is invariant under conjugation by elements in G. These groups are also known to be characterized as the kernel of some group homomorphism. Definition 2. A group G is solvable if there exists a chain of subgroups 1 = H0 ⊲ 2 −1 H1 ⊲ ::: ⊲ Hn = G such that Hi+1~Hi is abelian for i = 0; :::; n . Notice this means that solvable groups are groups that are constructed by extensions of abelian groups. Solvable groups will be the focus of this paper. Definition 3. The Frattini subgroup,Φ(G) is the intersection of all maximal proper subgroups of G. The Frattini subgroup always exists in finite groups and possesses many use- ful properties. For example, the Frattini subgroup is always normal. In the case of 2 2 2 3 2 3 D8 we have three maximal subgroups {1; s; r ; sr } , {1; r; r ; r }, and {1; rs; r ; sr }. 2 Therefore Φ(D8) = {1; r }. One property of the Frattini subgroup we will benefit from is that it is nilpotent. Definition 4. Let G be a finite group and p a prime. A Sylow p-subgroup of G is a subgroup P ≤ G such that SP S = pa is the full power of p dividing SGS. The set of all Sylow p-subgroups of G is denoted Sylp(G) Definition 5. Let G be finite and solvable and let π be any set of prime numbers. Hall's Theorem guarantees a subgroup H ≤ G with order divisible only by primes in π with SGS~SHS divisible by none of these primes. A subgroup H ≤ G satisfying these conditions is called a Hall π-subgroup of G. The Sylow and Hall subgroups will provide us a natural way to `split-up' groups into two subgroups using a free product. Definition 6. A group G is called nilpotent if there exists a chain of subgroups 1 = G0 ⊲ G1 ⊲ ::: ⊲ Gn = G such that Gi+1~Gi ≤ Z(G~Gi). Where Z(G~Gi) = {g ∈ G~GiSgx = xg for all x ∈ G~Gi}. 3 This formal definition can be replaced by another property that is equiva- lent to being nilpotent. A group G is nilpotent if and only if it can be written as the direct product of its Sylow subgroups for all primes p dividing SGS. Notice that because D8 is a finite p-group that it must be a nilpotent group. Definition 7. The Fitting subgroup of G, written as F (G) is the unique largest normal nilpotent subgroup of G. In the example of D8, we would have F (D8) = D8, because D8 is nilpotent. The Fitting subgroup will appear in relation to Gasch¨utz'Theorem which we will state in the next section. Definition 8. Let H1 and H2 be subgroups of G. We define the commutator of these groups to be −1 −1 [H1;H2] = ⟨h1 h2 h1h2Sh1 ∈ H1; h2 ∈ H2⟩: The commutator subgroup of G is the group [G; G] and denoted by G′. The commutator subgroup is the smallest normal subgroup such that the quotient group of the group by its commutator is abelian. It turns out that SG~G′S has a relation to the largest orbit size of a group action. In the case of D8 the com- mutator subgroup is {1; r2}.
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